Let $A$ be a compact convex set in $\mathbb R^n$. Let $y\in \mathbb R^n$ be an arbitrary point not belonging to $A$. Let $P$ be a hyper-plane which separates $A$ and $y$. Let $x$ be the projection of $y$ onto $P$.
We now recall the definition of gauge norm. Gauge norm of a point $z\in \mathbb R^n$ with respect to $A$ is defined by $$||z||_A=\inf \{t>0: z\in tA\}.$$
Here, we are interested in the comparison between gauge norm of $y$ and $x$. I intuitively observe that $$||y||_A\geq ||x||_A.$$
It looks quite trivial but I was unable to prove it. Am I missing something? Or it is just a trivial fact in convex geometry?
I don't think it's true if we allow any plane separating y and A, see the picture (assume that the origin coincides with the midpoint of the triangle's base). It is true though if you assume that the plane is orthogonal to the line connecting $y$ with its projection on (the closest point of) $A$.